evolution - Population genetics and the fitness probability distribution. Why is the arithmetic mean all we need?

When recording change in allele frequency in diploid, bi-allelic, infinite and panmixic population we usually use this kind of equation:

$\delta_p = \frac{p * q *( p (w11 - w12) + q * (w12 - w22))}{\bar{w}}$

$\bar{w} = p^2 * w11 + 2*p*q*w12+q^2*w22$

$\delta_p$ = change of $p$ (frequency of one of the allele) from one time step to another

$w11$ is the mean fitness of individuals of genotype 11. $p$ and $q$ are the allele frequencies.

The only indicators for the fitness distribution is the arithmetic mean. Why don't we include other indicator of the probability distribution of fitness? The skew, the sd, the median for examples. Could you argue why we don't need to care about the probability distribution of fitness of individuals with genotypes 11 (for example)? In other words, why is the mean fitness (=w11) a sufficient statistics?

I wouldn't be able to answer if one asks me:

1. Why don't you take the median instead of the arithmetic mean?"




2. Why don't you care about the variance, the skew (or any other moment) of your distribution?




3. What if the traits were not countinuous but discrete (sex is a discrete trait for example)?